A phase transition in Erdős-Barak random graphs

We study monotone paths in Erdős-Rényi random graphs on numbered vertices. Benjamini & Tzalik established a phase transition at $p=\frac{\log n}{n}$ for this model. We refine the critical value to $p=\frac{\log n-\log \log n}{n}$ and identify the critical window of order $\mathrm{\Theta }(1\mathrm{/}n)$.